Abstract:In polygonal billiard dynamics, the interplay between the piece-line regular and vertex-angle singular boundary
effects is related by mathematicians to the problem of integrability and by physicists to the problem of causality and
randomness. We approach to the controversial issue of vertex-splitting effects by employing the alternative deterministic
and stochastic schemes. The theoretical study is developed within the frameworks i) of the billiard-wall collision
statistics of orbits [1] and ii) their survival probability, simulated in closed [2] and weakly open [3] polygons,
respectively. The role of vertex-splitting effects in late-time relaxation dynamics in pseudo-integrable polygons is
revealed through the comparative analyses with the ballistic dynamics in the integrable circular billiard, and the
superdiffusive orbit relaxation in the chaotic Sinai billiards. In the multi-vertex polygons, the orbit-splitting events
participate, due to the high-order rotational symmetry of the boundary, in the surviving of circular-like (sliding) regular
orbits, be means of their modification in new irregular excitations (vortices). Having no topological analog in the
geometrically corresponding circular billiard, both kinds of the orbits are observed through the universal and non-universal
channels of relaxations, respectively.